ESTIMATION OF FINITE POPULATION MEAN WITH KNOWN COEFFICIENT OF VARIATION OF AN AUXILIARY CHARACTER H.P. Singh, R. Tailor
1.INTRODUCTION AND THE SUGGESTED ESTIMATOR
It is well known that the use of auxiliary information at the estimation stage provides efficient estimators of the parameter (s) of the study character y. When the population mean X of the auxiliary character x is known, a large number of estimators such as ratio, product and regression estimators and their modifica-tions, have been suggested by various authors. Das and Tripathi (1980) have ad-vocated that the coefficient of variation Cx of the auxiliary character x is also available in many practical situations. Keeping this in view, Sisodia and Dwivedi (1981), Singh and Upadhyaya (1986) and Pandey and Dubey (1988) have made the use of coefficient of variation Cx alongwith the population mean X in esti-mating the population mean Y of y.
Suppose n pairs (xi, yi ) (i =1,2,...,n) observations are taken on n units sam- pled from N population units using simple random sampling without replace-ment (SRSWOR) scheme. The classical ratio and product estimators for Y are respectively defined by ˆ R X Y y x § · ¨ ¸© ¹ (1) and ˆ P x Y y X § · ¨ ¸© ¹ (2) where 1 / n i i x
¦
x n, 1 / n i i y¦
y n and 1 / N i iX
¦
x N is the known populationmean of the auxiliary character x.
mean X of x is also known, Sisodia and Dwivedi (1981) suggested a ratio-type estimator for Y as ( ) ˆ ( ) x MR x X C Y y x C (3)
and Pandey and Dubey (1988) proposed a product – type estimator for Y as
( ) ˆ ( ) x MP x x C Y y X C (4)
Motivated by Rao and Mudholkar (1967) and Singh and Ruiz Espejo (2003), we suggest a ratio - product estimator for Y as
ˆ x (1 ) x MRP x x X C x C Y y x C X C D D ª § · § ·º « ¨ ¸ ¨ ¸» « © ¹ © ¹» ¬ ¼, (5)
where D is a suitably chosen scalar. We note that for D =1, YˆMRP reduces to the estimator YˆMR suggested by Sisodia and Dwivedi (1981) while for D = 0 it re-duces to the estimator YˆMP reported by Pandey and Dubey (1988).
2.BIAS OF YˆMRP
To obtain the bias of YˆMRP, we write
0 (1 ) y Y e 1 (1 ) x X e such that 0 1 2 2 0 2 2 1 2 0 1 ( ) ( ) 0 (1 ) ( ) (1 ) ( ) (1 ) ( ) y x x E e E e f E e C n f E e C n f E e e C C n U ½ ° ° °° ¾ ° ° ° °¿ (6)
where Cy Sy/ ,Y Cx Sx/X, U Syx/S Sx y, C UCy/Cx, 2 2 2 2 1 1 1 ( ) /( 1), ( ) /( 1), ( )( )/( 1). N N x i y i i i N xy i i i S x X N S y Y N S x X y Y N
¦
¦
¦
Expressing (5) in terms of e’s we have
1 0 1 1 ˆ (1 )[ (1 ) (1 )(1 )] MRP Y Y e D Te D Te , (7) where T X X( Cx)
We now assume that Te1 so that we may expand 1 (1Te1)1 as a series in powers of Te1. Expanding, multiplying out and retaining terms of e’s to the sec-ond degree, we obtain
2 2 0 1 1 0 1 0 1 0 1 ˆ [ (1 ) (1 )(1 )] MRP Y #Y D e Te T e Te e D e Te Te e or 2 2 0 1 0 1 1 0 1 1 ˆ (YMRP Y)#Y e¬ª Te Te e D T( e 2Te e 2Te )º¼ (8)
Taking expectations of both sides of (8), we obtain the bias of YˆMRP to order 0(n1) as 2 (1 ) ˆ ( MRP) f x[ ( 2 )] B Y Y C C C n T D T (9) which vanishes if /(2 ) C C D T (10)
Thus the estimator YˆMRP with D C/(2C is almost unbiased. We also T) note from (9) that the bias of YˆMRP is negligible if the sample size n is sufficiently large.
To the first degree of approximation, the biases of YˆR, ˆ P Y , YˆMR and ˆ MP Y are respectively given by 2 (1 ) ˆ ( R) f x(1 ) B Y Y C C n (11)
2 (1 ) ˆ ( P) x f B Y Y C C n (12) 2 (1 ) ˆ ( MR) f x( ) B Y Y C C n T T (13) 2 (1 ) ˆ ( MP) f x B Y Y C C n T (14) From (9) and (11) we note that
ˆ ˆ ( MRP) ( R) B Y B Y if {C ( 2 )}C (1 C) T D T i.e. if {1 (1 )} {1 (1 )} ( 2 ) ( 2 ) {1 (1 )} {1 (1 )} ( 2 ) ( 2 ) C C either C C C C or C C T T T T T T T T T T T T D D ½ ° ° ¾ ° ° ¿ (15)
From (9) and (12) we note that
ˆ ˆ ( MRP) ( P) B Y B Y if {C ( 2 )}C C T D T i.e. if (1 ) (1 ) ( 2 ) ( 2 ) (1 ) (1 ) ( 2 ) ( 2 ) C C either C C C C or C C T T T T T T T T T T T T D D T ½ ° ° ¾ ° ° ¿ (16)
From (9) and (13) it follows that
ˆ ˆ
( MRP) ( MR)
B Y B Y if {C D T( 2 )}C (T C)
Further from (9) and (14) we see that ˆ ˆ ( MRP) ( MP) B Y B Y if {C D T( 2 )}C C i.e. if 2 0 ( 2 ) 2 0 ( 2 ) C either C C or C D T D T ½ ° ° ¾ ° ° ¿ (18) 3.VARIANCE OF YˆMRP
Squaring both sides of (8) and neglecting terms of e’s having power greater than two we have
2 2 2 2
0 1 0 1
ˆ
(YMRP Y) Y e[ (1 2 ) {(1 2 )D T D Te 2e e }] (19)
Taking expectations both sides in (19), we get the variance of YˆMRP to the first degree of approximation as 2 2 2 (1 ) ˆ ( MRP) f [ y (1 2 ) x{(1 2 ) 2 }] V Y Y C C C n D T D T (20) which is minimized for
0 ( ) 2 C T D D T (say) (21)
Substitution of (21) in (5) yields the asymptotically optimum estimator (AOE) for Y as ( 0 ) ˆ ( ) ( ) 2 x x MRP x x X C x C y Y C C x C X C T T T ª § · § ·º « ¨ ¸ ¨ ¸» « © ¹ © ¹» ¬ ¼ (22)
Putting (21) in (9) and (20) we get the bias and variance of YˆMRP( 0 ) respectively as
( 0 ) (1 ) 2 ˆ ( ) ( )( ) 2 MRP x f B Y Y C C C n T T (23) and ( 0 ) (1 ) 2 2 ˆ ( MRP) f y(1 ) V Y S n U (24)
It is to be mentioned here that the variance of YˆMRP( 0 ) at (24) is same as that of the approximate variance of the usual linear regression estimator
ˆ( )
lr
y y E X x , where Eˆ is the sample regression coefficient of y on x.
Remark 3.1. In practice, with a good guess of ‘C’ obtained through pilot sur-veys, past data or experience gathered in due course of time, an optimum value of
D fairely close to its true value D0 can be obtained. This problem has been also discussed among others by Murthy (1967, pp. 96-99), Reddy (1978) and Sri-venkataramana and Tracy (1980). Further if a good guess of the interval contain-ing ‘C’ (i.e. C1 d dC C2) which is more realistic than a specific guess about C, can be made on theory, accumulated experience and/or a scatter diagram for at least a part of current data then it is also advisable to use the suggested estimator in practice.
4. EFFICIENCY COMPARISONS
It is well known under SRSWOR that
2 2 (1 ) ( ) f y V y Y C n (25)
and the variance of YˆR, YˆP, YˆMR and YˆMP to the first degree of approximation are respectively given by
2 2 2 (1 ) ˆ ( R) f [ y x(1 2 )] V Y Y C C C n (26) 2 2 2 (1 ) ˆ ( P) f [ y x(1 2 )] V Y Y C C C n (27) 2 2 2 (1 ) ˆ ( MR) [ y x( 2 )] f V Y Y C C C n T T (28)
2 2 2 (1 ) ˆ ( MP) [ y x( 2 )] f V Y Y C C C n T T (29) From (20) and (25) we have
2 2 (1 ) ˆ ( MRP) ( ) f x (1 2 ){ (1 2 ) 2 } V Y V y Y C C n T D T D which is negative if 1 1 2 2 1 1 2 2 C either C or D T D T ½ § · ¨ ¸° © ¹° ¾ § · ° ¨ ¸ ° © ¹ ¿ (30)
From (20) and (26) we have
2 2 (1 ) ˆ ˆ ( MRP) ( R) f x{(1 2 ) 1}{(1 2 ) 2 1} V Y V Y Y C C n D T D T which is negative if (1 ) 2 1 2 2 (1 ) 2 1 2 2 C either C or T T D T T T T D T T § ½· ¨© ¸°¹° ¾ § · ° ¨ ¸ ° © ¹ ¿ (31)
From (20) and (27) we have
2 2 (1 ) ˆ ˆ ( MRP) ( P) f x{(1 2 ) 1}{(1 2 ) 2 1} V Y V Y Y C C n D T D T which is negative if ( 1) 2 1 2 2 ( 1) ( 2 1) 2 2 C either C or T T D T T T T D T T ½ § · ¨ ¸ °° © ¹ ¾ ° °¿ (32)
From (20) and (28) we have
2 2 (1 ) ˆ ˆ ( MRP) ( MR) 4 f x ( 1)( ) V Y V Y Y C C n T D D T which is negative if
1 1 C either C or D T D T ½ °° ¾ ° °¿ (33)
Further from (20) and (29) we have
2 2 (1 ) ˆ ˆ ( MRP) ( MP) 4 f x { ( 1) } V Y V Y Y C C n DT T D which is negative if 0 1 1 0 C either C or D T D T ½ § · ¨ ¸ ° © ¹ ° ¾ § · ° ¨ ¸ ° © ¹ ¿ (34)
5. ESTIMATOR BASED ON ESTIMATED ‘OPTIMUM’
If exact or good guess of ‘C’ is not available, we can replace ‘C’ by the sample estimate Cˆ in (22) and get the estimator (based on estimated optimum) as
( 0 ) ˆˆ ( ˆ) ( ˆ) 2 x x MRP x x X C x C y Y C C x C X C T T T ª § · § ·º « ¨ ¸ ¨ ¸» « © ¹ © ¹» ¬ ¼ (35)
where Cˆ (sxy/ ){1/y XCx2} where, we recall, X and Cx are known, and
1 ( )( )/( 1) n xy i i i s
¦
x x y y n .To obtain the variance of YˆˆMRP( 0 ) we write
2
ˆ (1 )
C C e
with E C( )ˆ C o n( 1).
Expressing YˆˆMRP( 0 ) in terms of e s' we have
( 0 ) 1 0 2 1 2 1 ˆˆ (1 )[{ (1 )}(1 ) { (1 )}(1 )] 2 MRP Y Y e T C e Te T C e Te T
( 0 ) ( 0 ) 2 ˆ ˆ ˆ ˆ ( MRP) ( MRP ) V Y E Y Y 2 2 0 1 2 1 2 1 (1 ) [{ (1 )}(1 ) { (1 )}(1 )] 1 2 e Y E T C e Te T C e Te T § · ¨ ¸ © ¹ (36) Expanding the terms on the right hand side of (36) and neglecting power of '
e s that are greater than two we have
( 0 ) 2 2 2 2 2 2 0 1 0 0 1 1 ˆˆ ( MRP) ( ) ( 2 ) V Y Y E e C e Y E e C e e C e (1 ) 2[ 2y 2 x2 2 y x)] (1 ) 2y(1 2) f f Y C C C C C C S n U n U
which is same as that of YˆMRP( 0 ) i e V Y. . ( ˆMRP( 0 ) ) V Y( ˆˆMRP( 0 ) ). Thus it is established that the variance of the estimator YˆˆMRP( 0 ) in (35) based on the estimated optimum, to terms of order n1, is the same as that of YˆMRP( 0 ) in (22).
6.EMPIRICAL STUDY
To examine the merits of the suggested estimator we have considered five natural population data sets. The description of the population are given below. Population – I: Murthy (1967, p. 228) N= 80, y: Output n = 20, x: Fixed Capital 51.8264, 11.2646, y 0.3542, x 0.7507, Y X C C 0.9413, C 0.4441 f 0.25, 0.9375. U T ,
Population – II: Murthy (1967, p. 228)
N= 80, y: Output n = 20, x: Number of Workers 51.8264, 2.8513, y 0.3542, x 0.9484 Y X C C 0.9150, C 0.3417 f 0.25, 0.7504. U T ,
Population – III: Das (1988)
N= 278, y: Number of agricultural labourers for 1971 n = 30, x: Number of agricultural labourers for 1961
39.0680, 25.1110, y 1.4451, x 1.6198,
Y X C C
0.7213, C 0.6435 f 0.1079, 0.9394.
Population – IV: Steel and Torrie (1960, p. 282) N= 30, y: Log of leaf burn in secs
n = 6, x: Clorine percentage 0.6860, 0.8077, y 0.700123, x 0.7493, Y X C C 0.4996, C 0.3202 0.5188 f 0.20 U T , Population – V: Maddala (1977)
N= 16, y: Consumption per capita
n = 4, x: Deflated prices of veal
7.6375, 75.4313, y 0.2278, x 0.0986,
Y X C C
0.6823 C 1.5761 0.9987 f 0.25
U T
We have computed the ranges of D for which the proposed estimator YˆMRP is better than y , YˆR , YˆP , YˆMR and YˆMP , optimum value of D and common range of D and displayed in Table 1. Table 2 shows the percent relative efficien-cies of YˆMRP with respect to y , YˆR , YˆP, YˆMR and YˆMP.
TABLE 1
Range of D in which YˆMRP is better than y , Y ,ˆR Y ,ˆP YˆMR and YˆMP Popu-
lation Range of D in which YˆMRP is better than
Optimum value ofD y ˆ R Y YˆP YˆMR YˆMP D0 Common range of D in which ˆ MRP Y is better than y ,YˆR , ˆ P Y , YˆMR andYˆMP . I (0.50,0.9737) (0.4404,1.0333) (-0.033, 1.5070) (0.4737, 1.00) (0.00, 1.4737) 0.73685 (0.50, 0.9737) II (0.50,0.9554) (0.2891,1.1663) (-0.1663,1.6217) (0.4554, 1.00) (0.00, 1.4554) 0.72768 (0.50, 0.9554) III (0.50,1.1850) (0.6528,1.0323) (-0.0323,1.7173) (0.6850, 1.00) (0.00, 1.6850) 0.84251 (0.6850, 1.00) IV (-0.1172,0.50) (-1.0810,1.4638) (-0.4638,0.8466) (-0.6172, 1.00) (0.00, 0.3828) 0.19140 (0.00, 0.3828) V (-1.0782,0.50) (-1.5788,1.0007) (-0.577,-0.0007) (-1.5782, 1.00) (-0.5782, 0.00) -0.28908 (-0.5775,0.0007) TABLE 2
Percent relative efficiencies of YˆMRP( 0 ) or YˆˆMRP( 0 ) with respect to y ,
ˆ
R
Y , Y ,ˆP YˆMR andYˆMP
Population Percent relative efficiencies of YˆMRP with respect to:
y YˆR YˆP YˆMR YˆMP I 877.62 1318.18 * 1059.44 * II 614.40 2008.96 * 835.86 * III 208.46 133.29 * 122.94 * IV 133.26 * 249.90 * 112.80 V 187.37 * 111.91 * 111.88
Table 1 exhibits that there is enough scope of selecting the scalar ‘D ’ in YˆMRP
to get better estimators. It is observed that even if D slides away from its true optimum value, the efficiency of the suggested estimator YˆMRP can be increased considerably. Table 2 clearly indicates that the suggested estimator YˆMRP( 0 ) or YˆˆMRP( 0 )
is more efficient (with substantial gain) than the usual unbiased estimator y , clas-sical ratio estimator YˆR and product estimator YˆP , and the modified estimators
ˆ MR
Y and YˆMP suggested by Sisodia and Dwivedi (1981) and Pandey and Dubey
(1988) respectively. Thus the proposed estimator YˆˆMRP( 0 ) is to be preferred in
prac-tice.
7.CONCLUSION
This article is concerned with estimating the population mean Y of the study variate y using auxiliary information at the estimation stage. When the population mean X and coefficient of variation Cx of an auxiliary variable x is known, a class of estimators for estimating Y is suggested. ‘Optimum’ estimator in the class is identified with its approximate variance formula. Estimator based on es-timated optimum values is also proposed with its approximate variance formula. It is interesting to note that the estimators based on ‘optimum value’ and ‘esti-mated optimum value’ have the same approximate variance formula. Thus we conclude that the studies carried out in the present article can be used fruitfully even if the optimum values are not known. An empirical study is carried out to throw light on the performance of the suggested estimator over already existing estimators. Further empirical studies carried out in this article clearly reflect the usefulness of the proposed estimators in practice.
School of Studies in Statistics H.P. SINGH
Vikram University, Ujjain, India RITESH TAILOR
ACKNOWLEGDEMENTS
Authors are thankful to the referees for making valuable suggestion towards improving the presentation of the material.
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RIASSUNTO
Stima della media di un popolazione finita con coefficiente di variazione di un carattere ausiliario noto
Il contributo si occupa del problema della stima di una media di popolazione Y di una variabile oggetto di studio y utilizzando l’informazione sulla media di popolazione X e sul coefficiente di variazione Cx di un carattere ausiliario x. Viene suggerito uno stima- tore per il parametro Y e ne vengono studiate le proprietà nel contesto di singoli cam- pioni. Si dimostra che lo stimatore proposto, sotto alcune condizioni realistiche, è più ef- ficiente degli stimatori proposti da Sisodia e Dwivedi (1981) e da Pandey e Dubey (1988). Tramite una analisi empirica vengono esaminati i meriti dello stimatore costruito rispetto agli antagonisti.
SUMMARY
Estimation of finite population mean with known coefficient of variation of an auxiliary character
This paper deals with the problem of estimating population mean Y of the study vari-ate y using information on population mean X and coefficient of variation C of an x
auxiliary character x. We have suggested an estimator for Y and its properties are studied in the context of single sampling. It is shown that the proposed estimator is more effi-cient than Sisodia and Dwivedi (1981) estimator and Pandey and Dubey (1988) estimator under some realistic conditions. An empirical study is carried out to examine the merits of the constructed estimator over others.
